Spread of a Gaussian Wave Packet with Time

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A localized free particle can be represented by a Gaussian wave packet. This Demonstration shows the spreading of a Gaussian wave packet, considering the effects of varying particle mass and momentum and initial width of the wave packet. Choose the parameters: mass, initial width and momentum, and see the evolution of the wave packet with time.

Contributed by: Radhika Prasad and Sarbani Chatterjee  (January 2018)
Supervised by: S. N. Sandhya
(Miranda House, University of Delhi)
Open content licensed under CC BY-NC-SA


Snapshots


Details

A Gaussian wave packet at time can be represented by

,

where is the initial width of the wave packet and is the momentum.

At a later time , the wave packet evolves with

,

where is a constant, is the mass of the particle, and is a parameter that determines the corresponding width of the wave packet ( is the actual width). The parameter is given by

Clearly, the width increases with time , as the wave packet spreads. For simplicity, and have been set equal to unity.

The probability amplitude is a Gaussian function centered about the point .

The wave packet maintains a Gaussian shape, with changing centroid and width.

Snapshot 1: small initial width implies faster spread

Snapshot 2: initially broad wave packet spreads out more slowly

In the limiting case, of a wave packet initially equal to a Dirac delta function, it immediately transforms into an infinite plane wave.

Reference

[1] H. C. Verma, Quantum Physics, 2nd ed., Bhopura, Ghaziabad, India: Surya Publications, 2009.



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